## Three - dimensional Geometry

# Plane

- The equation of the plane passing through a point A(\vec{a}) whose normal vector is \vec{n} is \vec{r}\cdot \vec{n}=\vec{a}\cdot \vec{n}
- The equation of plane passing through a point A(x
_{1}, y_{1}, z_{1}) and a, b, c are the direction ratios of normal is a(x − x_{1}) + b(y − y_{1}) + c(z − z_{1}) = 0 - The equation of the plane which is at a distance d units from origin and whose normal vector is \vec{n} is \vec{r}\cdot \hat{n}=d where \hat{n}=\frac{\vec{n}}{|\vec{n}|}
- The equation of the plane which is at a distance d units from origin and a, b, c are direction ratios of normal is lx + my + nz = d

where l=\frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}, m=\frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}, n=\frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}} - The equation of the plane passing through the points A(x
_{1}, y_{1}, z_{1}), B(x_{2}, y_{2}, z_{2}) and C(x_{3}, y_{3}, z_{3}) is \begin{vmatrix}x-x_{1} & y-y_{1} & z-z_{1}\\ x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1}\\ x_{3}-x_{1} & y_{3}-y_{1} & z_{3}-z_{1} \end{vmatrix}=0 - The equation of the plane passing through the point (x
_{1}, y_{1}, z_{1}) and parallel to the lines whose directions are (a_{1}, b_{1}, c_{1}) and (a_{2}, b_{2}, c_{2}) is \begin{vmatrix}x-x_{1} & y-y_{1} & z-z_{1}\\ a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2} \end{vmatrix}=0 - The equation of the plane passig through the point (x
_{1}, y_{1}, z_{1}) and perpendicular to the two plane a_{1}x + b_{1}y + c_{1}z + d_{1}= 0, and a_{2}x + b_{2}y + c_{2}z + d_{2}= 0 is \begin{vmatrix}x-x_{1} & y-y_{1} & z-z_{1}\\ a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2} \end{vmatrix}=0 - The equation of the plane containing two lines \frac{x-x_{1}}{a_{1}}=\frac{y-y_{1}}{ b_{1}}=\frac{z-z_{1}}{c_{1}} and \frac{x-x_{2}}{a_{2}}=\frac{y-y_{2}}{ b_{2}}=\frac{z-z_{2}}{c_{2}} is \begin{vmatrix}x-x_{1} & y-y_{1} & z-z_{1}\\ a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2} \end{vmatrix}=0
- The equation of the plane passing through the points (x
_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}) and parallel to the line whose directions are (a_{1}, b_{1}, c_{1}) is \begin{vmatrix}x-x_{1} & y-y_{1} & z-z_{1}\\ x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1} \\ a_{1} & b_{1} & c_{1} \end{vmatrix}=0 - The equation of the plane passing through the points (x
_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}) and perpendicular to the plane a_{1}x + b_{1}y + c_{1}z + d_{1}= 0 is \begin{vmatrix}x-x_{1} & y-y_{1} & z-z_{1}\\ x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1} \\ a_{1} & b_{1} & c_{1} \end{vmatrix}=0 - The equation of the plane containing two parallel lines \frac{x-x_{1}}{a_{1}}=\frac{y-y_{1}}{ b_{1}}=\frac{z-z_{1}}{c_{1}} and \frac{x-x_{2}}{a_{1}}=\frac{y-y_{2}}{ b_{1}}=\frac{z-z_{2}}{c_{1}} is\begin{vmatrix}x-x_{1} & y-y_{1} & z-z_{1}\\ x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1} \\ a_{1} & b_{1} & c_{1} \end{vmatrix}=0
- The equation of the plane making intercepts a, b, c on the x, y, z - axis respectively is \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1
- The equation of the plane passing through the intersection of the two planes a
_{1}x + b_{1}y + c_{1}z + d_{1}= 0, and a_{2}x + b_{2}y + c_{2}z + d_{2}= 0 is (a_{1}x + b_{1}y + c_{1}z + d_{1}) + λ (a_{2}x + b_{2}y + c_{2}z + d_{2}) = 0 - The equation of the plane passing through the intersection of the two planes \vec{r}\cdot \vec{n}_{1}=d_{1} and \vec{r}\cdot \vec{n}_{2}=d_{2} is \vec{r}\cdot (\vec{n}_{1}+\lambda \vec{n}_{2})=d_{1}+\lambda d_{2}
- The area of the triangle formed by the plane \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1 with x-axis, y-axis is \frac{1}{2}|ab|

The area of the triangle formed by the plane \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1 with y-axis, z-axis is \frac{1}{2}|bc|

The area of the triangle formed by the plane \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1 with z-axis, x-axis is \frac{1}{2}|ca| - The equation of the plane passing through the point (x
_{1}, y_{1}, z_{1}) and parallel to the plane ax + by + cz + d = 0 is a(x − x_{1}) + b(y − y_{1}) + c(z − z_{1}) = 0 - The equation of xy-plane z = 0

The equation of yz-plane x = 0

The equation of zx-plane y = 0 - The equation of the plane passing through (x
_{1}, y_{1}, z_{1}) and parallel xy-plane is z = z_{1}

The equation of the plane passing through (x_{1}, y_{1}, z_{1}) and parallel yz-plane is x = x_{1}

The equation of the plane passing through (x_{1}, y_{1}, z_{1}) and parallel zx-plane is y = y_{1} - The equation of the palne midway between the planes ax + by + cz + d
_{1}= 0 and ax + by + cz + d_{2}= 0 is ax + by+cz\left(\frac{d_{1}+d_{2}}{2}\right)=0 - Equation of the plane parallel to x-axis is \frac{y}{b}+\frac{z}{c}=1

Equation of the plane parallel to y-axis is \frac{x}{a}+\frac{z}{c}=1

Equation of the plane parallel to z-axis is \frac{x}{a}+\frac{y}{b}=1 - The vector equation of the plane passing through a given point A(\vec{a}) and parallel to the vectors \vec{b},\vec{c} is [\vec{r}-\vec{a} \ \ \ \vec{b} \ \ \ \vec{c}] = 0
- The vector equation of the plane passing through origin and points \vec{b},\vec{c} is [\vec{r} \ \ \ \vec{b} \ \ \ \vec{c}]=0

### Part1: View the Topic in this video From 23:26 To 54:29

### Part2: View the Topic in this video From 01:30 To 47:42

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1. In the vector form, equation of a plane which is at a distance d from the origin, and \hat{n} is the unit vector normal to the plane through the origin is \vec{r}\cdot\hat{n}=d.

2. Equation of a plane which is at a distance of d from the origin and the direction cosines of the normal to the plane as l, m, n is lx + my + nz = d.

3. The equation of a plane through a point whose position vector is \vec{a} and perpendicular to the vector \vec{N} is (\vec{r}-\vec{a}).\vec{N}=0.

4. Equation of a plane perpendicular to a given line with direction ratios A, B, C and passing through a given point (x_{1}, y_{1}, z_{1}) is A(x − x_{1}) + B(y − y_{1}) + C(z − z_{1}) = 0

5. Equation of a plane passing through three non collinear points (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) is \begin{vmatrix}x-x_{1} & y-y_{1} & z-z_{1}\\x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1} \\x_{3}-x_{1} & y_{3}-y_{1} & z_{3}-z_{1} \end{vmatrix}=0

6. Vector equation of a plane that contains three non collinear points having position vectors \vec{a},\vec{b} \ and \ \vec{c} is(\vec{r}-\vec{a}).[(\vec{b}-\vec{a})\times (\vec{c}-\vec{a})]=0

7. Equation of a plane that cuts the coordinates axes at (a, 0, 0), (0, b, 0) and (0, 0, c) is \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1

8. Vector equation of a plane that passes through the intersection of planes \vec{r}\cdot \vec{n}_{1}=d_{1} and \vec{r}\cdot \vec{n}_{2}=d_{2} is \vec{r}\cdot \left(\vec{n}_{1}+\lambda \vec{n}_{2}\right)=d_{1}+\lambda d_{2}, where λ is any nonzero constant.

9. Vector equation of a plane that passes through the intersection of two given planes A_{1}x + B_{1}y + C_{1}z + D_{1} = 0 and A_{2}x + B_{2}y + C_{2}z + D_{2} = 0 is (A_{1}x + B_{1}y + C_{1}z + D_{1}) + λ(A_{2}x + B_{2}y + C_{2}z + D_{2}) = 0.